Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001615: Lattices ⟶ ℤ
Values
[1] => [[1],[]] => ([],1) => 0
[1,1] => [[1,1],[]] => ([],1) => 0
[2] => [[2],[]] => ([],1) => 0
[1,1,1] => [[1,1,1],[]] => ([],1) => 0
[1,2] => [[2,1],[]] => ([],1) => 0
[2,1] => [[2,2],[1]] => ([],1) => 0
[3] => [[3],[]] => ([],1) => 0
[1,1,1,1] => [[1,1,1,1],[]] => ([],1) => 0
[1,1,2] => [[2,1,1],[]] => ([],1) => 0
[1,2,1] => [[2,2,1],[1]] => ([(0,1)],2) => 1
[1,3] => [[3,1],[]] => ([],1) => 0
[2,1,1] => [[2,2,2],[1,1]] => ([],1) => 0
[2,2] => [[3,2],[1]] => ([(0,1)],2) => 1
[3,1] => [[3,3],[2]] => ([],1) => 0
[4] => [[4],[]] => ([],1) => 0
[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,2] => [[2,1,1,1],[]] => ([],1) => 0
[1,1,2,1] => [[2,2,1,1],[1]] => ([(0,1)],2) => 1
[1,1,3] => [[3,1,1],[]] => ([],1) => 0
[1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,1)],2) => 1
[1,2,2] => [[3,2,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,3,1] => [[3,3,1],[2]] => ([(0,1)],2) => 1
[1,4] => [[4,1],[]] => ([],1) => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([],1) => 0
[2,1,2] => [[3,2,2],[1,1]] => ([(0,1)],2) => 1
[2,2,1] => [[3,3,2],[2,1]] => ([(0,2),(2,1)],3) => 2
[2,3] => [[4,2],[1]] => ([(0,1)],2) => 1
[3,1,1] => [[3,3,3],[2,2]] => ([],1) => 0
[3,2] => [[4,3],[2]] => ([(0,1)],2) => 1
[4,1] => [[4,4],[3]] => ([],1) => 0
[5] => [[5],[]] => ([],1) => 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,2] => [[2,1,1,1,1],[]] => ([],1) => 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,1)],2) => 1
[1,1,1,3] => [[3,1,1,1],[]] => ([],1) => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,2,2] => [[3,2,1,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,1,3,1] => [[3,3,1,1],[2]] => ([(0,1)],2) => 1
[1,1,4] => [[4,1,1],[]] => ([],1) => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,1)],2) => 1
[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[1,2,3] => [[4,2,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,1)],2) => 1
[1,3,2] => [[4,3,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,1] => [[4,4,1],[3]] => ([(0,1)],2) => 1
[1,5] => [[5,1],[]] => ([],1) => 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => ([],1) => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => ([(0,1)],2) => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,3] => [[4,2,2],[1,1]] => ([(0,1)],2) => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,2),(2,1)],3) => 2
[2,2,2] => [[4,3,2],[2,1]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[2,3,1] => [[4,4,2],[3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4] => [[5,2],[1]] => ([(0,1)],2) => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([],1) => 0
[3,1,2] => [[4,3,3],[2,2]] => ([(0,1)],2) => 1
[3,2,1] => [[4,4,3],[3,2]] => ([(0,2),(2,1)],3) => 2
[3,3] => [[5,3],[2]] => ([(0,2),(2,1)],3) => 2
[4,1,1] => [[4,4,4],[3,3]] => ([],1) => 0
[4,2] => [[5,4],[3]] => ([(0,1)],2) => 1
[5,1] => [[5,5],[4]] => ([],1) => 0
[6] => [[6],[]] => ([],1) => 0
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => ([(0,1)],2) => 1
[1,1,1,1,3] => [[3,1,1,1,1],[]] => ([],1) => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,1)],2) => 1
[1,1,1,4] => [[4,1,1,1],[]] => ([],1) => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,2),(2,1)],3) => 2
[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,4,1] => [[4,4,1,1],[3]] => ([(0,1)],2) => 1
[1,1,5] => [[5,1,1],[]] => ([],1) => 0
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => ([(0,1)],2) => 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 4
[1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 3
[1,2,4] => [[5,2,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => ([(0,1)],2) => 1
[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2
[1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 3
[1,3,3] => [[5,3,1],[2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,1)],2) => 1
[1,4,2] => [[5,4,1],[3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,5,1] => [[5,5,1],[4]] => ([(0,1)],2) => 1
[1,6] => [[6,1],[]] => ([],1) => 0
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([],1) => 0
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => ([(0,1)],2) => 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,1)],2) => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 3
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Description
The number of join prime elements of a lattice.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Map
dominating sublattice
Description
Return the sublattice of the dominance order induced by the support of the expansion of the skew Schur function into Schur functions.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the subposet of the dominance order whose elements are the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a lattice.
The example $\lambda = (5^2,4^2,1)$ and $\mu=(3,2)$ shows that this lattice is not a sublattice of the dominance order.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the subposet of the dominance order whose elements are the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a lattice.
The example $\lambda = (5^2,4^2,1)$ and $\mu=(3,2)$ shows that this lattice is not a sublattice of the dominance order.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
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